AN EDUCATION star Carey Mulligan regrets dying her hair for Johnny Depp's PUBLIC ENEMIES - because the damaging chemicals she used cost her her prized locks.The Pride & Prejudice star thought her small role in the big budget feature would elevate her career, so she had no problems agreeing to dye her hair at the request of movie bosses. But the new look cost her dear and she admits the boyish hairdo she now sports is more necessity than fashion statement. She says, "My hair is a bit of a sore topic. I miss my hair. I did a tiny extra part in Public Enemies with Johnny Depp and I got really into it and they said, 'Will you dye your hair?' I said yes because I thought that would advance my career. "It (hair) turned to straw, so I just cut it off."